Determining the domain of the quotient \(q\)  in terms of the \(H\) -action on \(Y\)  is quite delicate, even though we can determine the domain of \(q_{\overline{L}}\)  using the Hilbert–Mumford criterion for the reductive group \(\overline{L}\) . If \(T\)  is a maximal torus of \(L\)  containing \(\lambda (\mathbb {G}_m)\) , one would like a description of \(\operatorname{dom}(q)\)  in terms of \(T\) -weights on \(Y\) , but passing to the \(U\) -quotient involves deleting some of these weights. Nevertheless, we can sandwich the domain of definition between sets admitting Hilbert–Mumford type descriptions in terms of weights as in Theorem REF   REF below. Moreover, under the assumption REF , we can explicitly determine the domain of \(q\) . In [1]}, a Hilbert–Mumford type statement is also stated without assuming REF .
